theorem
  IB is max-ideal implies for a holds a in IB iff not a` in IB
proof
  reconsider FB = IB.: as Filter of B.:;
  assume IB is max-ideal;
  then
A1: FB is being_ultrafilter by Th32;
  let a;
  a.:` = a` by Lm4;
  hence thesis by A1,FILTER_0:46;
end;
